DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020352 DYNAMICAL SYSTEMS Volume 41, Number 5, May 2021 pp. 2051{2070 CYLINDER ABSOLUTE GAMES ON SOLENOIDS L. Singhal Beijing International Center for Mathematical Research Peking University Beijing, 100 871, China Current address: Yau Mathematical Sciences Center Tsinghua University Beijing, 100 084, China (Communicated by Enrique Pujals) Abstract. Let A be any affine surjective endomorphism of a solenoid ΣP 1 over the circle S which is not an infinite-order translation of ΣP . We prove the existence of a cylinder absolute winning (CAW) subset F ⊆ ΣP with the property that for any x 2 F , the orbit closure fA`x j ` 2 Ng does not contain any periodic orbits. A measure µ on a metric space is said to be Federer if for all small enough balls around any generic point with respect to µ, the measure grows by at most some constant multiple on doubling the radius of the ball. The class of infinite solenoids considered in this paper provides, to the best of our knowledge, some of the early natural examples of non-Federer spaces where absolute games can be played and won. Dimension maximality and incompressibility of CAW sets is also discussed for a number of possibilities in addition to their winning nature for the games known from before. 1. Introduction. Let P be an (finite or infinite) ordered set of prime numbers with p1 < p2 < ··· and the restricted product space Y 0 XP := R × p2P Qp; Q0 where denotes that for each element x 2 XP , the entries xp are in the compact 1 ring Zp for all but finitely many p's. A P-solenoid over the unit circle S is the X quotient space ΣP := P=∆(R), where R = Z [ f 1=p j p 2 P g ] is a subring of Q embedded diagonally as the uniform lattice ∆(R) = (r)p2f1g[P j r 2 R ⊂ XP : 2020 Mathematics Subject Classification. Primary: 11J61; Secondary: 28A80, 37C45. Key words and phrases. Dynamical systems, Hausdorff dimension, Incompressible sets, Non- dense orbits, Schmidt's game. Parts of this work first appeared in a slightly different avatar in the author's PhD thesis submitted to the Tata Institute of Fundamental Research, Bombay in 2017. For a portion of that duration, financial support from CSIR, Government of India under SPM-07/858(0199)/2014- EMR-I is duly acknowledged. 2051 2052 L. SINGHAL We denote the quotient map XP ! ΣP to be Π. Solenoids are compact, connected metrizable abelian groups. Their higher topological-dimensional cousins have some- times been called \fractal versions of tori" [21, Abstract]. When P is a finite set of cardinality l − 1, the Hausdorff dimension of XP (and therefore of ΣP too) under the natural metric given by (4) is l. This also implies that the dimension is infinite when P is so, as the increasing sequence of finite product spaces associated with the finite truncations of P are isometrically embedded inside XP . The set of endomorphisms of ΣP is precisely the ring R whose elements act multiplicatively componentwise. An affine transformation A :ΣP ! ΣP is meant to denote the map x 7! (m=n)x + a (1) where m=n 2 Rnf0g and a 2 ΣP . It is well known that when A = m=n is a surjective endomorphism of the solenoid, it acts ergodically on ΣP iff m=n2 = f0; ±1g [23, Proposition 1.4]. We also learn from [4, Theorem 3.2] that for every compact group G, any semigroup of its affine transformations lying above an ergodic semigroup of surjective endomorphisms is ergodic as well. This gives us a sufficient condition for the transformation Ax = (m=n)x + a to be ergodic, namely that m=n 6= ±1. Ergodicity of the action guarantees that almost all orbits of A are dense in ΣP . However, just like [8], this work is concerned with understanding the complementary set. Given an affine transformation A, we would like to know how large is the set of points of ΣP whose A-orbits remain away from periodic B-orbits for all B 2 R n {±1g. When P is the set consisting of all the primes in N, the space ΣP is called the full solenoid over S1 with the field Q being its ring of endomorphisms. Let B be a non-zero rational number. The growth of the number of B-periodic orbits as a function of the period is determined by the entropy of the action on ΣP . The latter has been computed first in [12] and recovered in [17], where it was explained to be the sum of the Euclidean and the p-adic contributions. In fact, Lind and Ward [17] achieve it for all automorphisms of solenoids over higher-dimensional tori as well. We remark that each such epimorphism lifts uniquely to a homomorphism from XP to itself, which we shall continue to denote by the same rational number. For an affine transformation, we however have a choice involved in terms of a representa- tive for the translation part a. Let y 2 ΣP be arbitrary and A be an affine surjective transformation of ΣP as in (1) with either m=n 6= 1 or a 2 Π(∆(Q)). We intend to show that the set of points x 2 XP whose forward orbit under the map x 7! Ax maintains some positive −1 distance from the 1-uniformly discrete subset Π (fyg) ⊂ XP is cylinder absolute winning (CAW) in a similar sense as [11]. To begin with, one observes that the affine transformation A is a finite-order translation of the solenoid when m=n = 1 and a 2 Π(∆(Q)). The set of points of XP whose (forward) orbit avoids some neighbourhood of Π−1(y) is then the complement of some uniformly discrete subset of XP and is certainly CAW. This is also the case when m=n = −1 and A becomes an involution of the space ΣP . Once we have a statement as advertised early in the paragraph above, we can take intersection of countably many of these sets to conclude about A-orbits which avoid neighbourhoods of all periodic orbits of surjective endomorphisms. This strat- egy is in similar taste and builds upon the work of Dani [8] on orbits of semisimple toral automorphisms. CYLINDER ABSOLUTE GAMES ON SOLENOIDS 2053 Our setup has two players in which one of them (Alice) will be blocking open cylinder subsets of XP at every stage of a two-player game. To elaborate, one such cylinder is given by ( Q Q 0 R × j<i Qpj × B(xi; pi") × j>iQpj if i > 0 and C(x; "; i) := Q0 (2) B(x0;") × j>0 Qpj otherwise, where x = (x0; : : : ; xi;:::) 2 XP . For us, B(xi; r) will always be the set of points in Qpi whose distance from xi is strictly less than r while B(xi; r) will also include those whose distance from xi is exactly r. We explain this game in x 3 after a brief tour of some of its older and related versions. The aim is to prove the following statement in this paper: 1 Theorem 1.1. Let ΣP be a solenoid over the circle S and fAj : x 7! (mj=nj)x + aj j j 2 Ng be any countable family of affine surjective endomorphisms of ΣP such that 1. none of the Aj's is a translation of ΣP of infinite order, and 2. the collection of rational numbers fmj=njg lying below the family fAjg j2N j2N belong to some finite ring extension of Z. Then, there exists a cylinder absolute winning subset F ⊆ ΣP such that for any k x 2 F and j 2 N, the orbit closure Aj x j k 2 N contains no periodic B-orbit for all B 2 R n {±1g. This is done in x 4. Note that the first condition in our hypothesis can possibly be relaxed in terms of some independence criterion on the co-ordinate entries of the translation vector aj. However, Haar-almost all infinite-order translations will lead to all orbits being dense. The second condition is a technicality of the proof and will be interesting to explore further. In the last section, we illustrate as to how information about the existence of winning strategies for Alice can be used to infer something about the f-dimensional Hausdorff measure of F . We also discuss the strong winning and incompressible nature of CAW subsets when P is finite. 1.1. Comparison with the work of Weil. It is plausible that some of our results given here may also be obtained from the more general framework discussed in [22]. We take some time to expound the import of his main result. Let X; d be a proper metric space (i.e. all closed balls are compact) and X a closed subset of X.A contraction for us would be a map from the half-open interval (0; 1] to the class of non-empty compact subsets of X satisfying (t1) ⊆ (t2) for all 0 < t1 < t2. Consider a family of subsets Rλ ⊆ X j λ 2 Λ which are called resonant sets and a family of contractions f λ j λ 2 Λg, indexed by some (same) countable set Λ. It is further required that Rλ ⊆ λ(t) for all λ 2 Λ and t > 0. This data is written in a concise form as F = (Λ;Rλ; λ). The set of badly approximable points in S with respect to the family F is defined as ( ) [ BAX (F) := x 2 X j 9c = c(x) > 0 such that x2 = λ(c) : (3) λ2Λ Next, each Rλ is assigned a height hλ with infλ hλ > 0.